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A100157
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Structured rhombic dodecahedral numbers (vertex structure 9).
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21
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1, 14, 55, 140, 285, 506, 819, 1240, 1785, 2470, 3311, 4324, 5525, 6930, 8555, 10416, 12529, 14910, 17575, 20540, 23821, 27434, 31395, 35720, 40425, 45526, 51039, 56980, 63365, 70210, 77531, 85344
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OFFSET
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1,2
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COMMENTS
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Also structured triakis octahedral numbers (vertex structure 9) (Cf. A100171 = alternate vertex); and structured heptagonal anti-prism numbers (Cf. A100185 = structured anti-prisms).
If Y is a 2-subset of a 2n-set X then, for n>=2, a(n-1) is the number of 4-subsets of X intersecting Y. - Milan Janjic, Nov 18 2007
Let M(2n-1) be a (2n-1)x(2n-1) matrix whose (i,j)-entry equals i^2/(i^2+sqrt(-1)) if i=j and equals 1 otherwise. Then a(n) equals (-1)^(n+1) times the real part of prod(k^2+sqrt(-1),k=1...2n-1) times the determinant of M(2n-1). - John M. Campbell, Sep 07 2011
The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link). a(n)= Cat(n,4), so enumerates the number of (n+1)-gon partitions of a (4*(n-1)+2)-gon. Analogous series are A000326 (k=3) and A234043 (k=5). Also, a(n)= A006918(4n+1) = A008610(4n+1) = A053307(4n+1) with offset=0. - Tom Copeland, Oct 05 2014
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REFERENCES
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Jolley, Summation of Series, Dover (1961).
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LINKS
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FORMULA
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a(n) = (16*n^3-12*n^2+2*n)/6.
sum_{n>=1} 1/(24*a(n)) = Pi/8-log(2)/2 = 0.046125491418751.. [Jolley eq. 251]
G.f. x*(1+10*x+5*x^2) / (x-1)^4 . - R. J. Mathar, Oct 03 2011
a(n) = binomial(2n+1,3) + binomial(2n,3). - John Molokach, Jul 10 2013
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EXAMPLE
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For n=4, sum( (4+i)^2, i=-3..3 ) = (4-3)^2+(4-2)^2+(4-1)^2+(4-0)^2+(4+1)^2+(4+2)^2+(4+3)^2 = 140 = a(4). - Bruno Berselli, Jul 24 2014
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MAPLE
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with(combstruct):ZL:=[st, {st=Prod(left, right), left=Set(U, card=r), right=Set(U, card=r), U=Sequence(Z, card>=1)}, unlabeled]: subs(r=1, stack): seq(count(subs(r=2, ZL), size=m*4), m=1..32) ; # Zerinvary Lajos, Jan 02 2008
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PROG
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CROSSREFS
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Cf. A005915 = alternate vertex; A100145 for more on structured polyhedral numbers.
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KEYWORD
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easy,nonn
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AUTHOR
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James A. Record (james.record(AT)gmail.com), Nov 07 2004
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STATUS
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approved
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